Science > Physics > Nonrenormalization vs Renormalization 10: Gamma Analyzed By PI
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Science > Physics |
| User: |
"OsherD" |
| Date: |
10 Apr 2006 07:56:51 PM |
| Object: |
Nonrenormalization vs Renormalization 10: Gamma Analyzed By PI |
From Osher Doctorow
The gamma function already derives from Probable Influence/Causation
(PI) via PIA maximum entropy of the gamma family of distributions and
other asymmetric nonnegative real line distributions, but some further
analysis yields some additional combinatorics.
Recall that for positive integer n, the gamma function G(n) is:
1) G(n) = (n - 1)! = 1 * 2 * 3 *** * n (using * as times, that is to
say multiplication)
The multiplication in (1) can be "converted" to addition in PI in the
same way that conditional probability y/x converts to 1 + y - x in PI
(the 1 has the role of insuring that the result is a probability, that
is to say between 0 and 1, and although the 1 derives from the
mathematics the key thing is the change from division to subtraction,
which for multiplication corresponds to a change to addition):
2) Sum(i) = 1 + 2 + 3 + ... + n = n(n + 1)/2
where Sum(i) is the sum of consecutive positive integers i from 1 to n.
Equation (2) is well known in algebra and calculus, and retains the
idea of finite counting (which is a positive integer scenario) across
addition, not just multiplication as in (1). The important point is
the sum 1 + 2 + 3 + ... + n in (2), not so much the n(n + 1)/2
expression although it facilitates computation. Notice that n(n +
1)/2 only has one product of "variables", n times n + 1 or (n/2) times
n + 1.
For the sum of square consecutive positive integers from 1^2 to n^2 the
result is the well known n(n + 1)(2n +1)/6, and for cubes it is [n(n +
1)/2]^2, which again are well known equations.
Readers can try to see what happens with 1 - 2 + 3 - 4 + 5 ..., which
has the difficulty that depending on how terms are rearranged it yields
different sums. This is the well known rearrangement difficulty of
nonconvergent infinite series. Could one define convergent subseries
with appropriate associative rules?
Osher Doctorow
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| User: "Amid" |
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| Title: Re: Nonrenormalization vs Renormalization 10: Gamma Analyzed By PI |
10 Apr 2006 08:58:50 PM |
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"OsherD" <> wrote in message
news:1144717011.064672.27310@g10g2000cwb.googlegroups.com...
From Osher Doctorow
The Hamas function already
has NOTHING to do with
Probable Influence/Causation
(PI) via PIA maximum entropy of the Hamas family of distributions and
other asymmetric nonnegative real line distributions, but some further
rectile analysis yields some additional combinatorics.
Recall that for positive integer n, the Hamas function G(n) is:
1) G(n) = (n - 1)! = 1 * 2 * 3 *** * n
you mean to *(n-1) (a good learning lesson for you!)
(using * as times, that is to
say multiplication)
The multiplication in (1) can be "converted" to addition in PI in the
same way that conditional probability y/x converts to 1 + y - x in PI
y / x = 1 + y - x => y = x +x*y - x^2
Or y = x*(1-x)/(1-x) = x
So x = y and y = x
(the 1 has the role of insuring that the result is a probability, that
is to say between 0 and 1, and although the 1 derives from the
mathematics the key thing is the change from division to subtraction,
which for multiplication corresponds to a change to addition):
2) Sum(i) = 1 + 2 + 3 + ... + n = n(n + 1)/2
where Sum(i) is the sum of consecutive positive integers i from 1 to n.
should be Sum(x) where x =1,n
this has nothing to do with Gamma Analisys !!
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Nonrenormalization vs Renormalization 10: Gamma Analyzed By PI |
11 Apr 2006 12:36:33 AM |
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From Osher Doctorow
Amid from palistine.com, whose profile/options list 2 postings total to
the internet of which his reply above is one, typed:
The Hamas function already
This is presumably supposed to be a quote from me from my previous
post, but this doesn't appear anywhere in my previous post. He goes on
with similar unreal remarks. Anyone interested in a "Palestinian"
version of Alice in Wonderland should read his post and then rethink
their support of trolls posing as underdogs (underrabbits?).
Osher
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