When is it appropriate to entitle a paper "A note on..." or "On the ..."? I rarely find modern research papers (on mathematics) that are less than 5 pages long. However, recently I came across a couple of mathematical research papers from the 1960/1970's that were very short (only 2-4 pages long). The authors of both papers solved very specific problems, and stopped writing (I guess) as soon as they were done. This made me realize that I very much like short papers! When possible, I will strive to do the same with my own future papers.
Sometimes one finds short papers that are entitled "A note on...". Since I am first of all not a native English speaker and also only a junior mathematician, I would like to know what you think about my questions/thoughts:
1) Is it a good (or bad?) idea to try to let the title of the paper reflect the fact that it is short? Are there any standard ways of doing this?
2) A paper which is entitled "A note on...", is it expected to be short? How short? Can a 50-pages long paper be "A note on..." or would that not be customary?
3) At least to me "A note on..." could give the impression that the paper is a survey article where no new material is presented, but I guess this need not be the case. When is it appropriate to entitle a paper "A note on..."?
4) When is it appropriate to entitle a paper "On the ..."?
 A: Some journals explicitly frown upon titles of that form. For example, the AMS Bulletin (strangely enough, I only found this information in the back matter of the printed/PDF version of the journal, but not directly on the website) states (emphasis original):

The first page must consist of a short descriptive title ... The descriptive title should be short but informative: useless or vague phrases such as "some remarks about" or "concerning" should be avoided.

A: Historically, it was quite common to start titles with "on," much more so than it is today. Many classic works of science have such titles, such as
"On the Origin of Species," by Charles Darwin;
"On Growth and Form," by D'Arcy Wentworth Thompson;
"De rerum natura," by Lucretius;
"De revolutionibus orbium coelestium," by Nicolaus Copernicus.
A: I feel the answer is NEVER.  You must describe the content of the article, not the length.  Some journals publish notes separately from regular papers, and often even encourage their submission by offering speedy refereeing and publication (even the Annals encourages "short", i.e. under 20 pp. papers).  Anyhow, if your paper is published as a note it will have "Note" written on it anyway, so no need to be redundant.  
More generally, you should emphasize not the length but the content.  If you prove that all tennis balls are white make the title "All tennis balls are white".  If you prove that some tennis balls are white, title your note "On white tennis balls", or "New examples of white tennis balls" or whatever.  If your note is a new simple proof, and this is what you want to emphasize, make the title "Short proof that all tennis balls are white".  If there was a conjecture that all tennis balls were white and you found a counterexample, use "Not all tennis balls are white".  If you study further color properties of white tennis balls, use "A remark on white tennis balls".  You see the idea.  
On the other hand, if you wrote a survey, it important to emphasize that, regardless whether it's long or short.  That's because this is a property of the content and style of presentation.  For example, "A survey on white tennis balls" or "White tennis balls, a survey in colored pictures", etc.  In fact, if your title is "A short survey on tennis balls colors", that would mean that your survey is short in content, as in "brief, incomplete", rather than in length - an important info for the reader to know.  
A: If you are writing it for online distribution and otherwise don't care, name it as you wish.
If you think it will be read and used by others, then a couple of common sense rules apply:
1) Don't leave anything very important out of the title.
2) Anything important you leave out of the title must be put in the abstract or introductory summary.
3) Anything left out of the title or the abstract or the introductory summary, you must put in the article.
4) Anything left out of the above should have a reference clearly mentioned in the bibliography.
5) Whenever possible, use common sense to tell you what to leave out.
Otherwise, this will probably undergo some sort of editorial and peer review, in which case your choice of title may not be used.  Regarding your suggestions, I think you should take pity on the researchers to come and have a descriptive but not over lengthy title.
There may be reasons to start a title with "On" or "A Note".  One reason against it is that your paper will be hard to find in an alphabetic listing of titles.
I am writing an article for public distribution with a working title beginning "Adventures in".  Join now and (try to) beat the rush.
Gerhard "Ask Me About System Design" Paseman, 2011.11.16
