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From: OsherD on 31 May 2010 10:06
From Osher Doctorow

Once again disavowing any explicit mention of Probable Causation/
Influence (PI) by the Indian researchers to be discussed, their work
actually relates PI to orbits of finite abelian (commutative) groups.
Their paper is:

1) "Degenerations and orbits in finite abelian groups," by Kunal Dutta
and Amritanshu Prasad (both Institute of Mathematical Sciences Chennai
India), 14 pages, arXiv: 1005.5222 v1 [math.GR] 28 May 2010.

Here I focus on the two "underlying" equations cited by Dutta and
Prasad from previous researchers G. A. Miller (1905) and
Schwqachhoffer & Stroppel (1999), respectively:

2) |G_(L, p) \ A_(L, p) | = (PRODUCT) (Li - Li+1 + 1), PRODUCT for i
= 1 to m-1.

3) left hand side of (2) = sum sum tau_ik PRODUCT(tau_i,j - tau_i,j+1
- 1), PRODUCT for j = 1 to k-1.

Here G_(L, p) is the automorphism group of A_(L, p) which is in turn Z/
p^(L1) Z + ... + Z/p^Lm Z where L = (L1 > = ... > = Lm) is a non-
increasing sequence of positive integers, also known as a partition.
Every finite abelian p-group is isomorphic to an A(L, p). The tau_i,
j are actually tau_i with i having a sub-index (lower index) j where
tau1 < tau2 < ... < tau_t are the Li without repetitions, that is to
say the distinct positive integers in the partition L.

Group orbits are intuitively like orbits in physics and engineering
and astrophysics/astronomy, although group orbits are more general.
See Wikipedia's "Group orbit," "Group action," "Symmetry group", and
corresponding papers by Wolfram. The orbit can be regarded as a list
of what each element is mapped into or permuted into, so that if
element a is mapped to element b which is mapped to element c, for
example, then we could write a --> b --> c to describe the orbit.

Readers will recognize Li - Li+1 + 1 and similarly for tau_i as
Probable Causation/Influence (PI) or respectively its negative, 1 + y
- x, provided that the Li and tau_i are normalized into [0, 1]. In
PI, y < = x.

Osher Doctorow

From: OsherD on 31 May 2010 10:19
From Osher Doctorow

There is no "q" in the name Schwachhoffer.

By the way, as Wikipedia and Wolfram point out, the additive group of
real numbers t acts on the phase space of classical mechanics and
dynamical systems in such a way that a state x of the system at a
certain time is described t seconds later by:

1) tx (more usually t dot x) for t > 0

and for t seconds earlier by t dot x for t < 0. Here t dot x or tx
is the result of the group action of group G on set X, which is a
binary function G x X --> X, that is (g, x) --> g dot x or gx that is
associative and such that ex = x for e the identity in G and x any
element of X.

Osher Doctorow
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