Natural Science Forum / Physics / General Physics / August 2007

Symmetric and Antisymmetric Tensors relate both to Generalized
Alternating-Sign Matrices and other properties of Probable Causation/
Influence (PI), while Dirac Mass relates to "PI-Maximum Entropy", and
Entanglement relates to and both Traces and Eigenvalues which are
invariants of determinants under PI operations of +/-.

For the latest on Entanglement, look at "Entanglement in disordered
systems at criticality," Imre Varga of Fizikai Intezet Budapest
Hungary and U. Marburg Germany, and Jose ANtonio Mendez-Bermudez of U.
Autonoma de Puebla Mexico, 4 pages, arXiv: 0708.3682 (2-column pages
for each page).  This paper presents the above relationships quite
clearly although of course the authors are not familiar with PI.

For the latest on the Dirac Mass, read "On the Dirac-Infeld-Plebanski
delta function," Oscar Rosas-Ortiz of CINEVESTAV Mexico, 15 pages,
arXiv: 0705.0376 v1 [quant-ph] 2 May 2007, which does similarly for
the above relationships with the same qualification.

Also for the latest on Dirac Mass and its relationship to the Lotka-
Volterra equations which are generalizations of the Riccati
Differential Equation of PI, see "Dirac concentrations in Lotka-
Volterra parabolic PDEs," Benoit Perlhanno and Guy Barles (resp. U
Pierre et Marie Curie Paris and CNRS U. Francois Rabelais Tours
France, 22 pages, arXiv: 0708.3720 v1 [math.AP] 28 Aug 2007 (I have to
check the spellings of the authors' names because I wrote them fast).

Readers may still be somewhat confused about Dirac Mass without
looking at Wikipedia and Wolfram online, including "Dirac delta
function" Wikipedia and "Delta function" Wolfram.

For Symmetric and Antisymmetric Tensors, start with Wikipedia's
"Symmetric tensor".  I'll try to continue this shortly, except to
mention that antisymmetric Tijk = -Tjik and symmetric Tijk = Tjik for
example (i, j, k subscripts) are equivalent to the remarkable
generalized PI equations:

1) P(-Tijk --> Tjik) = 1
2) P(Tijk --> Tjik) = 1

respectively.   The -Tijk in (1) is a generalization of A in P(A-->B)
of PI or P ' (A-->B) of PI.   Readers can derive these equations from
using Tijk + Tjik = 0 and Tijk - Tjik = 0 respectively, adding "tensor
1" to both sides.

Osher Doctorow