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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 ..."?

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"A note" does suggest it is short (in principle you could have a 50 page note, but that would be an eccentric title). People may assume a note is not very important, so I'd hesitate to use that title unless either experts will consider the paper obviously important or you have other, better papers and don't care if this one sounds less important. It doesn't suggest a survey paper to me. Overall, I'd use "A note on..." or "On the..." only if that is the clearest title you can think of (and I think it often won't be). –  Henry Cohn Nov 17 '11 at 0:49
@David: I would say it implies that the author(s) thought it was not 'very important', which is something different entirely. –  quid Nov 17 '11 at 1:00
I tend to entitle my papers "(a note/remark) on (whatever problem)" when I cannot really solve the problem but think that what I can say is directly related to it and interesting enough to be published. A typical example is "A remark on the Mahler conjecture: local minimality of the unit cube ". The title says it all: we could not prove the actual conjecture but we still did something not completely trivial, which somewhat justified our going public with it. The length of the paper has nothing whatsoever to do with my choice of such titles. –  fedja Nov 17 '11 at 2:07
Regarding question 4, I would advise never starting a paper with the word "On." Just omit the word entirely and the title will be fine, and probably better, without it. (The only exception I can think of is D. E. Evans's paper "On $\cal O_n$.") –  Timothy Chow Nov 17 '11 at 3:29
I would start the title of a paper with "On [something]" if the [something] is already known. For example, a paper introducing generalized choral sequences would be titled "Generalized Choral Sequences" but a paper on some of its properties would be titled "On Generalized Choral Sequences." –  Joel Reyes Noche Nov 17 '11 at 3:49
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4 Answers

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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.

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I love your examples! Except that tennis balls are usually lime green or yellow ;-). –  Greg Graviton Nov 17 '11 at 12:59
Greg, it sounds like you need to write “On Pak’s tennis-ball conjecture”. –  L Spice Nov 17 '11 at 14:54
But, if he write it with his lime green/yellow slant he needs to hope for a 'modern' referee; the classical color for a tennis ball is white! –  quid Nov 17 '11 at 16:38
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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.

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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.

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Kudos on citing Lucretius! There's an excellent new translation by David Slavitt. –  Barry Cipra Nov 17 '11 at 23:08
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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

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I have never had a referee or editor suggest a change of title for one of my papers and I have never, in turn, suggested a title change to a paper I've handled as referee or editor, nor have I ever heard of such a thing. Has this ever happened to you? –  Felipe Voloch Nov 17 '11 at 6:16
It has not happened to me, but I can well imagine it happening to me because of my penchant for double entendre; I believe it is rare in mathematics but not totally unknown for a junior mathematician to make a poor initial choice in title. Gerhard "Has A Spotty Publication Record" Paseman, 2011.11.16 –  Gerhard Paseman Nov 17 '11 at 6:55
My paper "Cohomology detects failures of the axiom of choice" was originally titled "On the cohomology of discrete spaces" (or something very similar), but the referee objected. –  Andreas Blass Nov 17 '11 at 13:27
@Gerhard: "If you think it will be read and used by others, then..." Don't we all think (hope) so? Otherwise, what is the point of writing it? :) –  Johan Öinert Nov 17 '11 at 20:55
I have written some things down purely for my benefit, and made them available to others as a convenient form of service rather than with the expectation that it would be read (otherwise I might have to go to the trouble of writing for an audience rather than writing for myself). Gerhard "Does Practice Writing For Others" Paseman, 2011.11.17 –  Gerhard Paseman Nov 17 '11 at 21:44
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