Prev: Who shaves the barber who only shaves those who do not shave themselves?
Next: An ocean blanketed one-third of Mars about 3.5 billion years ago, a new study suggests
From: Osher Doctorow on 19 Jun 2010 18:29
From Osher Doctorow

PI Games are simply Games with maximum Probable Causation/Influence
(PI):

1) P(A-->B) = 1 + y - x, y = P(AB), x = P(A)

or:

2) P ' (A-->B) = 1 + y - x, y = P(B), x = P(A), y < = x

Both occur when y = x, in which case P and P ' are 1.

The equation y = x can also be generalized to:

3) y^2 - yx = 0 = y(y - x)

Let us consider the following matrix analog of this:

4) A^2 - AX = 0, where A and X are diagonal matrices.

Define:

5) A = diag(a1, a2) (matrix with a11 = a1, a22 = a2, a12 = 0 = a21)

The following is a solution of (4):

6) X = diag(a2, a1).

To prove this, just compute A^2 for A a 2x2 diagonal matrix diag(a1,
a2), which turns out to be diag(a1^2, a2^2), and then compute AX for X
= diag(a2, a1), yielding diag(a1, a2)diag(a2, a1) = diag(a1a2,
a1a2). Equating A^2 and AX yields, or preferably subtracting A^2 -
AX and using the fact that y = x (with a1 = y, a2 = x or vice versa):

7) a1^2 = a1a2, so a1 = a2, and a2^2 = a1a2, so a2 = a1, analogously
to x = y or y^2 = yx.

Notice that:

8) diag(a1, a2) = PALINDROME of diag(a2, a1) (a generalization of the
"palindrome", for example see this online under Wikipedia or Wolfram).

Osher Doctorow
From: Osher Doctorow on 19 Jun 2010 20:02
From Osher Doctorow

Technically, a "Palindrome" should involve the equality of a word or
string of symbols and its "reverse", so the word "Reversed" of a word
is more accurate here than Palindrome, although in genetics some
modifications occur.

Palindromes occur in DNA sequences and protein description and are
under study by various researchers, while "reversed word" occurs also
in braid groups, Grobner basis in semigroups, noncommutative algebras,
etc.).

arXiv has 19 papers under the keyword "Palindrome", 2 in 2010, 3 in
2009, 23 papers under "de Bruijn" which is related to Palindromes (6
in 2010, 5 in 2009), while "Word reversing, word reversed, reversed
word" occur in 3 papers and "Reversing" in 90 of whih only 6 seem
relevant (one in 2009, one in 2007, one in 2006, 2 in 2004, one in
2003).

One author in the above categories has 25 papers in arXiv, Patrick
Dehornoy of France.

Osher Doctorow
 | 
Pages: 1
Prev: Who shaves the barber who only shaves those who do not shave themselves?
Next: An ocean blanketed one-third of Mars about 3.5 billion years ago, a new study suggests