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This small page on normal forms has displaced the earlier and much more extensive page on matrix normal forms. Given their relative content, perhaps this page could be renamed to "normal forms (classification)" and "normal forms" be used for disambiguation. --Jjg 15:03, 19 April 2012 (CEST)

Or maybe this page itself is (or will be) an (extended) disambiguation page with links to detailed pages "normal form (for X)"? On Wikipedia in such cases one writes like this:
Matrices of linear maps between different linear spaces
Main article: Normal form (for matrices)
Such matrices are rectangular...
--Boris Tsirelson 17:05, 19 April 2012 (CEST)
This page is still under construction: I planned to complete it in the nearest future. By the way: I understand that it is a bad idea to "save page" when it is only partially written, but I have no idea on how to protect the work between sessions. The right solution would be to prepare the "complete" version in an off-line editor capable for expanding all EoM "macros", but I am unaware of any such editor under Windows (sorry, I know that's a bad taste ;-). The initial (rich) page is still available as Normal form (for matrices), and I plan to write a separate page for other types of normal forms in Dynamical systems, singularities, Lagrangian/Legendrian singularities, Hamiltonian systems etc. My idea was to collect under the common header "normal forms" different flavors of this notion with appropriate links to specific pages. Sergei Yakovenko 17:28, 19 April 2012 (CEST)
Yes, there is a nice way to do it: create and use a sandbox! I did; here are two examples: User:Boris Tsirelson/sandbox1, User:Boris Tsirelson/sandbox2. --Boris Tsirelson 21:10, 19 April 2012 (CEST)
I think some disambiguation would be helpful to the casual user (I'll admit that if I was looking for matrix normal forms then I'd head to the entry marked "normal forms" and be disappointed there were no matrices mentioned there)
--Jjg 17:36, 19 April 2012 (CEST)

Negative results

$\newcommand{\M}{\mathscr M}$ As was noted, the normal form of an object $M\in\M$ is a "selected representative" from the equivalence class $[M]$, usually possessing some nice properties. The set of all these "representatives" intersect each equivalence class exactly once; such set is called a transversal (for the given equivalence relation). Existence of a transversal is ensured by the axiom of choice for arbitrary equivalence relation on arbitrary set. However, a transversal in general is far from being nice. For example, consider the equivalence relation "$x-y$ is rational" for real numbers $x,y$. Its transversal (so-called Vitali set) cannot be Lebesgue measurable!

Typically, the set $\M$, endowed with its natural σ-algebra, is a standard Borel space, and the set $\{(x,y)\in\M\times\M:x\sim y\}$ is a Borel subset of $\M\times\M$; this case is well-known as a "Borel equivalence relation". Still, existence of a Borel transversal is not guaranteed (for an example, use the Vitali set again).

Existence of Borel transversals and related properties of equivalence relations are investigated in descriptive set theory. According to [K, Sect. 4], a lot of work in this area is philosophically motivated by problems of classification of objects up to some equivalence. A number of negative results are available. They show that in many cases, classification by a Borel transversal is impossible, and moreover, much weaker kinds of classification are also impossible.

[K] Alexander S. Kechris, "New directions in descriptive set theory", Bull. Symb. Logic 5 (1999), 161–174. Zbl 0933.03057


Do you like to include this section (near the end)? --Boris Tsirelson 17:02, 24 April 2012 (CEST)

Boris, it is almost "on purpose" that I avoided formal definitions in this page. Whatever equivalence relation you start with, very soon you reach the degeneracy level where no meaningful classification is possible, so the equivalence has to be "relaxed" to have any chance to continue. All the way around, while many classifications are partitions into orbits of suitable group actions, quite a few are not (cf. with the logical normal forms, but one can also think about Groebner bases etc. or your example from the set theory). As a result, I decided to subdivide the "normal form" cluster into "subject areas" the way it understand by the community. This is not always clear-cut, e.g., the page on the normal forms for matrices should include matrices of maps, matrices of operators, matrices of quadratic forms etc. In the "nonlinear" cases the corresponding object belong to different classes and are treated in detail on separate pages.
I suggest that the normal forms which do not arise from the "singular" classification problems, be mentioned at the disambiguation part near the top of the page and addressed either in separate pages, or as sections in the corresponding topical articles, like DST. Sergei Yakovenko 06:54, 25 April 2012 (CEST)
Well, maybe some day I'll create "Borel equivalence relation" article, and then you'll mention it here in the style you like. --Boris Tsirelson 08:14, 25 April 2012 (CEST)
How to Cite This Entry:
Normal form. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Normal_form&oldid=25337