Re: [math-fun] X^2, ln(X), repeat...
"Huddleston, Scott" <scott.huddleston@intel.com> wrote:
. . .
I think I understand what you're saying. There's a unique fixed point which I can find by alternating between e^X and the negative branch of sqrt(X). It's equal to about -0.703467. There's a unique cycle of length 2 which I can find by alternating between e^X and alternating between the positive and negative branches of sqrt(X). It's equal to about -1.298510. There are two cycles of length 3, whose values are about -1.482929 and -1.840801. (In each case, I'm listing the smallest number that appears in the cycle.) The number of cycles for lengths 1 through 7 are 1,1,2,3,6,9,18, which may be A066313. (Of all the OEIS sequences that begin with those numbers, that's the only one that mentions necklaces.) Trying to go further gets me precision loss. Here's a table: 1 -0.703467 2 -1.298510 3 -1.482929 3 -1.840801 4 -1.402786 4 -2.142474 4 -2.441350 5 -1.428439 5 -1.467677 5 -1.928954 5 -2.004383 5 -2.981728 5 -3.287670 6 -1.405661 6 -1.419140 6 -1.882384 6 -2.047197 6 -2.076965 6 -2.639042 6 -2.709934 6 -4.614471 6 -4.935768 7 -1.422375 7 -1.427156 7 -1.470046 7 -1.477157 7 -1.899349 7 -1.911679 7 -2.021488 7 -2.031539 7 -2.101821 7 -2.121867 7 -2.519123 7 -2.556653 7 -2.788948 7 -2.817744 7 -3.771299 7 -3.847934 7 -10.852741 7 -10.935144 These cycles are all repellers. The process bounces from one to to another pseudo-randomly forever. It might be interesting to graph all these numbers. Are they dense? Are they a cantor dust? What do they look like extended into the complex plane? Or would that just be asking for trouble, since the log function would also become multivalued? Is this really the first time anyone has studied this process/ sequence/system? How can that be answered, anyway? It's very hard to do a Google search on anything mathematical.
Here is a diagram of the values of f(x) = ln(x*x), showing the fixed and period-2 points: http://spacefilling.blogspot.com/ Kerry On Sun, Mar 24, 2013 at 8:16 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
"Huddleston, Scott" <scott.huddleston@intel.com> wrote:
. . .
I think I understand what you're saying. There's a unique fixed point which I can find by alternating between e^X and the negative branch of sqrt(X). It's equal to about -0.703467. There's a unique cycle of length 2 which I can find by alternating between e^X and alternating between the positive and negative branches of sqrt(X). It's equal to about -1.298510. There are two cycles of length 3, whose values are about -1.482929 and -1.840801. (In each case, I'm listing the smallest number that appears in the cycle.) The number of cycles for lengths 1 through 7 are 1,1,2,3,6,9,18, which may be A066313. (Of all the OEIS sequences that begin with those numbers, that's the only one that mentions necklaces.) Trying to go further gets me precision loss.
Here's a table:
1 -0.703467
2 -1.298510
3 -1.482929 3 -1.840801
4 -1.402786 4 -2.142474 4 -2.441350
5 -1.428439 5 -1.467677 5 -1.928954 5 -2.004383 5 -2.981728 5 -3.287670
6 -1.405661 6 -1.419140 6 -1.882384 6 -2.047197 6 -2.076965 6 -2.639042 6 -2.709934 6 -4.614471 6 -4.935768
7 -1.422375 7 -1.427156 7 -1.470046 7 -1.477157 7 -1.899349 7 -1.911679 7 -2.021488 7 -2.031539 7 -2.101821 7 -2.121867 7 -2.519123 7 -2.556653 7 -2.788948 7 -2.817744 7 -3.771299 7 -3.847934 7 -10.852741 7 -10.935144
These cycles are all repellers. The process bounces from one to to another pseudo-randomly forever.
It might be interesting to graph all these numbers. Are they dense? Are they a cantor dust? What do they look like extended into the complex plane? Or would that just be asking for trouble, since the log function would also become multivalued?
Is this really the first time anyone has studied this process/ sequence/system? How can that be answered, anyway? It's very hard to do a Google search on anything mathematical.
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participants (2)
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Keith F. Lynch -
Kerry Mitchell