Math paper with no "Conclusions"

Question

Is it acceptable for a math paper to have no concluding section? The structure is currently as follows:

1. Introduction (known results + the paper's main results as 3 theorems)
2. Preliminaries (proper definitions of all the things and recalling of results used in the proofs)
3. Proofs (actually, several lemmas and proofs of them, and of the theorems)
4. Open problems

It's quite a short paper. I thought that since the Introduction is actually containing all the important stuff, making a Conclusions section would be only repeating the same stuff again. However, I'm not sure if this is considered a poor style in maths.

Answer 1

The majority of papers in pure math have no concluding section. In fact, such papers most commonly end with the last line of the proof of the last main result (or the last lemma needed for the last main result) of the paper.

I think that I have never seen a pure math paper that has a concluding section in the sense of other academic papers, i.e., whose sole purpose is to summarize what has already happened. Some math papers have a final section which is more forward-looking. Your current structure has as a final section "Open Problems". That is as close to a concluding section as I've ever seen.

In fact though even this much is often not really appreciated or desired (by editors and referees anyway): many of my earlier papers contained a substantial "open problems / further work" section. Once I got the complaint that it seemed that I was trying to stake out territory (and there was some truth to that, I suppose). More than once I've been told that including too much speculation can be "embarrassing" when the truth comes to light. (In one case, I recorded computations that suggested a surprising conjecture. The referee said that I could keep that in if I wanted, but strongly implied that if I were more experienced I would know not to do this. I left it in, and a few years later my first student proved a theorem confirming these calculations.) In general if you include too much material in a math paper which is not directly used to prove the theorems of the paper, then referees start to wonder whether there is enough content in the paper to justify taking up all that valuable journal space: they could after all publish another theorem and proof instead. (And let's be fair: "they" have a point.)

If you're relatively young and inexperienced and hoping for best results on the rapid publication of your work in strong journals, I would stick pretty mercilessly to the format: (i) strong introduction motivating your work and explaining clearly the value added both in the results themselves and the techniques of proof and (ii) the rest of the paper contains careful proofs of all the results, in a very clear, linear, easy to follow fashion, e.g. "Section A.B: Proof of Lemma C".

Answer 2

Plenty of papers in math don't have a conclusion, or even a separate "open problems" section at the end - in fact, I think this is the norm.

Some specific examples: http://arxiv.org/pdf/1503.05803v1.pdf, http://arxiv.org/pdf/1503.05884v1.pdf, and http://arxiv.org/pdf/1503.05880v1.pdf - currently the first papers listed on the arxiv in logic, number theory, and algebraic geometry - have no conclusions.

Although, it is also by no means unheard of: http://arxiv.org/pdf/1503.05799v1.pdf and http://arxiv.org/abs/1503.05902 - the first papers in the "commutative algebra" and "algebraic topology" sections - do have (very short) conclusion-y sections.

Answer 3

In applied math (at least in my subfield), most papers do have a conclusion, and it's almost always a useless repetition of things already stated in the introduction (sometimes word-for-word). Please don't follow that pattern!

I try to use a conclusion only if there are general observations or discussion that I wish to include and that would not make sense in the introduction. This is rarely the case.