OEIS autoreply problem or Yahoo problem?
Jonathan Post
jvospost3 at gmail.com
Sat Feb 10 22:14:24 CET 2007
I iterate Jaap's "Congrats!"
But should I cut-and-paste the emails I sent about the 2 form
submissions that you may not have gotten, here (seqfans), or via
gmail to njas alone, or what?
As another text, I just submitted a sequence via form from a Linux box
rather than the former Windows 98 machine, and yahoo mailed a backup
to myself. My backups are usually via yahoo and gmail edresses for
redundancy. I'll observe what happens. The new one (in which I just
noticed the typo "provfes" for "proves"), with examples, OEIS xrefs,
and comments on its applicability to John Holland's 1976 book on the
genetic algorithm:
[backup copy]
Subject: NEW SEQUENCE FROM Jonathan Vos Post
%I A000001
%S A000001 2, 2, 3, 5, 7, 11, 17, 29, 38, 57, 86, 194,
291, 437, 656, 985, 1477, 2216, 3325
%N A000001 Floor((3^n-1)/(2^n-1)).
%C A000001 Ratio of the number of "schema" to
"chromosomes" in John Holland's model of simulated
evolution by natural selection. Suppose that there are
two alleles for every gene. That is, the simulated
chromosome is a binary string of length N. The
population is a fixed-size database of P of these
binary strings, each of which has an associated
fitness as a real value in the range [0,1]. We are
interested in what patterns there are in the
population. So we use "don't care" which I will
hereafter symbolize as d, as a metacharacter as used
in substring searches. So the set of possible schema
(patterns) is {(0,1,d)}^N. Let's take the small case
of N = 3. Then there are (2^3)-1 = 7 possible
chromosomes:
000, 001, 010, 011, 100, 101, 110, 111. There are
(3^3)-1 = 26 possible schema:
000, 00d, 0d0, d00, 0dd, d0d, ddd;
001, 0d1, d01, dd1;
010, 01d, d10, d1d;
011, d11;
100, 10d, 1d0, 1dd;
101, 1d1;
110, 11d;
111.
If there is no "d" in the string, then the pattern is
instantiated only by that string. That is, the schema
"010" has only the string "010" as an instance.
If there is one d in a schema, then it is instantiated
by any string where that d is substituted by either a
0 or a 1. For example, the string 00d is instantiated
by any of the set {000, 001}. If there is more than
one d, again one considers all substitutions. For
example, the string d0d is instantiated by any of the
set {000, 001, 100, 101} which is to say anything goes
so long as there is a 0 in the middle bit. The
remarkable theorem of Holland, which he rigorously
defines and provfes, is that evolution by natural
selection is implicitly and in parallel going on with
the superspace of schema in the larger schema fitness
landscape, and that follows the same statistical
difference equations (which in the limit become
differential equations) that define evolution by
natural selection on the population of strings.
%D A000001 John H. Holland, Adaptation in Natural and
Artificial Systems
An Introductory Analysis with Applications to Biology,
Control, and Artificial Intelligence, Ann Arbor:
University of Michigan Press, 1976; MIT Press, April
1992.
%F A000001 Floor(A024023(n)/A000225(n)).
%e A000001 n...(3^n-1)/(2^n-1)...a(n)
1...2.................2
2...2.66666667........2
3...3.71428571........3
4...5.33333333........5
5...7.80645161........7
6...11.5555556.......11
7...17.2125984.......17
8...25.7254902.......25
9...38.5166341.......38
10..57.7204301.......57
It is coincidence that the first 7 values are primes.
%Y A000001 Cf. A000225, A024023.
%O A000001 1
%K A000001 ,easy,more,nonn,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com),
Feb 10 2007
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