Talk:Paneitz operator

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GJMS operator
After rebuilding this article the article GJMS operator should be merged with this one as it is just a generalization.  franklin  15:37, 22 December 2009 (UTC)


 * On second thought, I definitely disagree with a merge from GJMS here. The GJMS operators are extremely important in conformal geometry in their own right.  The "just a generalization" argument doesn't wash with me.  Rings are "just a generalization" of the integers, but the two things have their own articles.  Sławomir Biały  (talk) 16:13, 22 December 2009 (UTC)
 * OK, but that is just your opinion. Don't think your self an authority and please don't remove the merge tag until consensus is arrived.  franklin   16:29, 22 December 2009 (UTC)
 * Also the ring-integer example doesn't quite fit. Just compare lengths of articles. —Preceding unsigned comment added by Franklin.vp (talk • contribs) 16:31, 22 December 2009 (UTC)
 * What would ask for an independent article? What makes Rings an independent article of Integers? It is a generalization that brings substantial new content, other particular versions of the general concept are also important topics independently of the particular one. The particular one has its own substantial content, having properties and uses far from being shared by all other cases or the general one. Agree that maybe the tag is not in the right article but either GJMS operator should be merged into this one or thisone into the other.  franklin   16:41, 22 December 2009 (UTC)

Yes, GJMS operators are substantially different from the Paneitz operators. In particular, the GJMS operators are naturally thought of as a hierarchy of differential operators coming from the ambient construction. While the Paneitz operators are in fact special cases, much more analysis has been done on them (e.g., positivity, eigenvalue problems, etc.) that a separate article could exist discussing these. The GJMS operators, particular the so-called "critical" GJMS operators, in turn are vital in the definition of the obstruction tensor and Q-curvature. So while one thing is a special case of the other, the two things are analyzed from completely different points of view, and have often very different kinds of results associated with them. Are you convinced? Sławomir Biały (talk) 16:49, 22 December 2009 (UTC)
 * Since I seem to be the only one around with enough expertise to make this determination, I will remove the merge tag in 24 hours unless another editor (not Franklin) objects, or Franklin presents an argument that is actually based in the literature and not just his own opinion that generalizations do not deserve separate articles on Wikipedia. Sławomir Biały  (talk) 17:18, 22 December 2009 (UTC)
 * It is tiresome to deal with your constant petulance and ignorance. For its own nature, it is necessary no expertise to manage to understand and edit any theme and article in Wikipedia. You thing too much of your self. Pity. Read the papers in the topic and you will see no mention of Paneitz's operator is made besides as being a particular case, or for dealing with the more descriptive computations that can be made due to the more simple technical situation. It is enough for me. I am not going to teach you anymore. Let it be your will two reidiculous stubs instead of a solid and consistent presentation. Bye bye, from now one these pages are unwatched for me.  franklin   18:56, 22 December 2009 (UTC)
 * I'm confused. Earlier you said, "It is a generalization that brings substantial new content, other particular versions of the general concept are also important topics independently of the particular one. The particular one has its own substantial content, having properties and uses far from being shared by all other cases or the general one." That sounds like an argument for keeping the two articles separate. Why would you want to merge the articles if you believe that there is substantial content to the special case of the Paneitz operator that is not true of the GJMS operator? Ozob (talk) 19:42, 22 December 2009 (UTC)
 * The Ring and the integers, were the subject of that statement.  franklin   19:54, 22 December 2009 (UTC)
 * So, how is this any different? Isn't it true that the Paneitz operator has special properties not shared by the general case? Ozob (talk) 00:21, 23 December 2009 (UTC)
 * The short answer is yes, especially from the PDE point of view. Also, the above claim that "you will see no mention of Paneitz's operator is made besides as being a particular case" is utterly unfounded.  Indeed, I have actually read some of these papers, and know most of the people, although it has been a number of years since I worked in this area.  I honestly have to doubt whether Franklin has done so, as he apparently claims to have.  But, if it's just a numbers game, here is a scholar search that should convince anyone not wishing to delve any deeper.   Sławomir Biały  (talk) 03:18, 23 December 2009 (UTC)