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I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar.

A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing with graduation from PHD. I do enjoy research, however the pressure to publish makes me be agitated and not quite, for I haven't publish any paper. I find, sometimes, doing research and to publish are contradictory.

Here is my question: How to write a good mathematical paper? Could anybody give me some suggestions?

Thanks ahead.

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It appears that you have done research and produced some results. Writing a paper should be easy once you have something to write about. You should write done your results and proofs as clearly as possible. Try not to get bogged down by details, and consult your advisor to determine what details an expert should be able fill in themselves. Check your spelling and grammar. Talk to advisor about the known journals of your area and those likely to accept your paper. The important thing is to consult your advisor, but since you have results, I think the hardest part is already done. –  William Aug 11 '12 at 1:38
Thanks William for your careful comment. The terrible thing is that I have not good results to write up to now. –  Paul Aug 11 '12 at 1:45
Don't do the classic textbook approach of having your main statement and the preceeding lemmas be a total secret until the reader get to the specific page. The most negative comment I recieved on the only thing I have ever written close to a mathematical paper was that I started out too "heavy". I was told it was better to have an abstract (which should only be a sentence or two) and then a relatively short section explaining basic ideas in a way that wouldn't be considered a wall of text or an overload of definitions and constructions. –  Arthur Aug 11 '12 at 1:48
You have to have a result first. –  user2468 Aug 11 '12 at 3:44
Having just refereed my first paper, I'll try to say a few of meaningful things. (1) Don't obfuscate with formally correct notation where a general idea -- simply expressible in English with perhaps a few mathematical symbols -- will suffice. (2) Be consistent with notations/conventions. (3) If your proof involves a long, tedious, technical component, break it up into segments and explain what it is you are attempting to do in each segment. (4) Remember that while after two years of intense study everything seems natural and clear, it might not be for someone seeing it for the first time. –  Arthur Fischer Aug 11 '12 at 7:07
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4 Answers

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As someone who is currently working on my first mathematical paper, I've found this guide from MIT to be very helpful. It covers both writing a clear and precise paper in general as well as the specific challenges presented by a mathematical paper. It's also fun to read! For example, the author likes to illustrate common mistakes within the text. One of my favorites is:

Don’t string adjectives together, especially if they are really nouns. Many high quality pure mathematics original research journal article sentences illustrate this problem.

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Thanks for your answer and your paper. The paper seems very helpful for me. I'v downloaded it:) –  Paul Aug 11 '12 at 1:58
+1 for the intelligent sentence. Made me laugh this morning! –  Gottfried Helms Aug 12 '12 at 8:34
it is funny)))) –  Seyhmus Güngören Aug 23 '12 at 8:54
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There are some notes on Mathematical Writing from a course taught by Knuth. They are quite extensive; I've only read the first few pages and those were already quite helpful to me, but there are also notes from guest lectures by various people, e.g. Wilf and Halmos.

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You say you haven't published a paper yet. Then perhaps this would be useful to you: How to Write Your First Paper by Steven G. Krantz (Notices of the AMS, December 2007, pp. 1507-1511).

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I have several pet peeves when I'm reading a paper, online article or blog which involves mathematics.

So if you are not constrained by space, and writing to the general public (and not for an expert community) try to avoid these:

  • Starting from an already simplified form: Even if it's tempting to remove a term from the both sides of the equation right away, I would like to see the formula or law used with all the substituted values in full glory and unchanged, and do the simplification in the next step.

  • Doing too many symbol folding in one step: when the author solves an equation, substitutes the resulting expression to another, solves it too, integrates it and factors it in a single step and shows the result only, and even not saying anything about how they get that result, and is even dare to write that it's left as an exercise for the reader...

  • Omitting the arguments of functions: using only $f$ to refer the value of $f(x)$. By brain decodes these argumentless variables as an arbitrary constant I can substitute any number to, then I wonder why they get that weird expression when they differentiate it using $x$. Differential equations and physics derivations involving speed and acceleration are guilty of this all the time...

Mathematicians should not strive to confuse their audiences.

(Feel free to add your own recurring annoyances, when reading papers...)

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I think a reason for why a lot of detail is left out of math papers is that when you're submitting to a journal, you want to keep the content as succinct and clear as possible. All details cannot be shown. Partly because it would bore a large part of the audience and partly because there just isn't space for it. I'm writing a paper now and if I showed every possible detail of all proofs, it would probably stretch to 20 pages. That isn't so bad, but the details are just not that enlightening whereas the general framework is. It's better to highlight the important and downplay the unimportant. –  Cameron Williams May 6 '13 at 15:06
If you focus too much on the details, the reader can get lost and not know what the important ideas are. You can easily obscure whatever beautiful result you have by cramming too much detail down the reader's throat. –  Cameron Williams May 6 '13 at 15:08
"it's left as an exercise for the reader" - algebra can be good for the reader's soul. :) –  J. M. May 6 '13 at 15:32
It is a subjective call (actually many of them in a row) to decide what details really are so obvious that the intended audience can easily fill them in; and this should be done when appropriate. That being said, there are far more journal articles with too few details than there are with too many details. The author should do the work once, rather than each reader doing it on his or her own. –  Greg Martin Apr 14 at 20:58
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