Re: Dimensional Analysis Via PI Versus Factor/Cluster/Scaling/PCA

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 15:21:20 +0000 (UTC)

On 29 Dec 04 02:07:18 -0500 (EST), Osher Doctorow wrote:
>variables are algebraically processed so to speak by the Buckingham
>PI Theorem, and if they've been selected correctly their functional
>equation emerges perfectly! If something is slightly "off" in the
conclude by mentioning "Fine structure constants in n-dimension-
>al physical spaces through dimensional analysis," by Fabricio
>Casrejos, Jaime F. Villas da Rocha, and Roberto Moreira Xavier,
>U. do Estado do Rio de Janeiro, arXiv:physics/0309040 v1 7 Sep 2003
>and "Analisi dimensionale: due interessanti applicazioni" by Germano
>D'Abramo (U. Roma) arXiv:physics/0306042 v1 5 Jun 2003, as well as

The name should be Casarejos, not Casrejos.

Also, the number of "independent" dimensionless products is under
quite general conditions equal to the number of variables in the
problem (for example, energy and momentum) minus the number of
dimensions, where dimensions used in physics and engineering are
typically mass (M), length (L), time (T), temperature (theta),
(electric) charge (Q). This is from the well known Buckingham PI
Theorem (PI in the sense of product, not Probable Influence, although
in the title of this thread PI refers to Probable Influence).

Readers who are interested in economics applications of dimensional
analysis should look up de Jong or De Jong as author on the internet
or in their research library, while those interested in population
applications can look up J. E. Bruno, O. Doctorow, and C. H. Kappner's
paper in a 1981 volume of Socioeconomic Planning Sciences (the latter
article contains a typographical error in an equation). Astrophysics
dimensional analysis has been published in a volume by Kurosh to my
recollection. Schepartz or Shepartz has likewise published a volume
on biological and medical dimensional analysis. The Franklin Insti-
tute has a volume or number entirely devoted to dimensional analysis.
Bluman and Kumei's Springer-Verlag volume (1989) Symmetries and
Differential Equations relates dimensional analysis to Lie Group
and Lie Algebra point transformations - note that their volume and
Bluman and Kumei's papers referenced therein played important roles
in Lie Group/Lie Algebra and similar analysis of ordinary and partial
differential equations.

Osher Doctorow