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The "research portion" of my paper is mostly done, but I'm having trouble putting together my research. (Keep in mind this is at the undergraduate level). It's not so much that I don't know how to organize my thoughts; in fact, I've written a very detailed outline. I seem to have a two questions I'm wrestling with (I apologize if this is a tad long-winded)

  1. The monotonous proof-theorem-proof-theorem style that essentially consists of a list of facts with no exposition seems to be most accepted. But will this really make my paper stand out? When I think about texts with proof-theorem-proof-theorem, I think of the color gray. If I try to be clever and witty, it could certainly backfire on me, but such papers are the most memorable papers I've ever read!

    So what should the tone/style of my paper be?

  2. I'm struggling very much with how detailed my paper should be. By no means am I writing a textbook, but I need to show the audience (namely admission committees) that I know what I'm talking about. On the other hand, I have a deadline! There's always this looming thought in the back of my head that says "But you can't leave that out!" when I debate if I should skip something just to speed up the process -- it's as if I have a hoarder living in my head. I have this same issue with considering how much background/prerequisite math I should include for the reader -- it is confusing, to say the least, writing to an audience that is more knowledgeable than the author. This is certainly very difficult for me to figure out.

    So how detailed should my paper be?

I'd really like to hear feedback from professors and/or admission committees members. But, of course, feedback from anyone with research experience is appreciated!

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    Although I think many junior researchers would have this type of question, the standard answer is Talk to your advisor.. I upvote it because the question shows that the OP is working very hard on his paper. – scaaahu Apr 3 '13 at 6:15
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    the audience (namely admission committees) — Unless you're planning to submit your paper to the Journal of Admissions Committees, that is not your audience. – JeffE Apr 3 '13 at 8:51
  • @JeffE: One might say it's his "target" audience at least. – aeismail Apr 3 '13 at 9:36
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    @aeismail: Nope. The best way to impress admissions committees with a paper is to ignore them completely and write the paper for the research community. If you try to make yourself look smart, your paper will suffer. – JeffE Apr 4 '13 at 1:00
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    @Mark: The process is intimidating for everyone! – JeffE Apr 4 '13 at 1:01
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Apologies for stealing scaaahu's comment into this answer, but I wanted to stress that it is really the only possible answer to this question: ask your advisor!

Getting a hang for the appropriate level of detail is a very tricky thing, and only comes with experience. Also with being clever and witty; if you can pull it off, great, but if you can't, then it distracts from the content of your article. There are no general rules for this, you need advice from a more experienced mentor.

If I were you, I would write up a quick draft without stressing too much over these issues, and go over it carefully with your advisor; they would be able to tell you whether your humour is working, or where you need more or less detail.

That said, a few general tips:

  1. Look at the papers you cite; if they all treat some theory as standard or well-known ("By class field theory....") then you probably can too. Same goes for tone.

  2. Often you can save space and omit details by rephrasing your results in a way that suggests how the proof should go. For example, let's say I have a lemma about a linear operator that gets proven by choosing a basis and then something straightforward; I might then state the lemma in terms of the matrix, so it would be more obvious to the reader how to complete the proof.

  3. (and the most important, imo) Engaging papers are not engaging because of witty one-liners, they are typically engaging because a great deal of effort went into organizing them well. The key is to ensure the reader always knows where they are, where they are going, and how they're getting there. This is not at all easy to do, but if you practice writing with this in mind early on, it will serve dividends later.

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    Very glad my comment is taken into a very good answer. – scaaahu Apr 3 '13 at 9:34
  • +1 for your #3. Putting the finishing touches on my first major paper and this has been the hardest part. – Mike A. May 15 '14 at 14:45
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My current field is not mathematics, but I am certainly not a real stranger to the field as I did my diploma (undergrad + masters) education on applied mathematics. Anyways here's my two cents:

  1. The regular structure and tone of research articles is pretty much the same; unless there is a good reason you stick to IMRaD (Intro, Mat&Met, Results and Discussion). Unless there is a good reason, you avoid passive tone, and changes in tense etc. This structure has the benefit of being familiar and intuitive to practically anyone out there. The "RaD" part is typically what's most important.

    What's different in your case is due to the fact that research om mathematics is more inductive than deductive. What that means is that you have to make sure your introduction is watertight, all your assumptions are valid. Since the value of your results depend on what you have built them on, the introduction becomes much more important than a "normal" introduction.

    Regarding the tone or style; the reason why some people can write witty is most likely because the same person has written many more articles, perhaps even book chapters. The experience really shines through in academic writing. Besides an established/respected professor isn't likely to be judged for humor in writing, the same way a junior scientist would be.

    Alternatively the person in question might be "witty" in his/her personality, and that wittiness passes on to the writing. As a young, aspiring academic you should be more worried about getting it right than getting it pretty/witty.

  2. It's usually hard to decide on that sort of stuff without having any more information. How much did you write? How much work are you presenting in that paper? It's usually a good idea to stick to the upper limit for the paper. Any journal (in my experience) will give you an upper limit on number of words/characters/pages for manuscript submission. I can only imagine admission committees would do the same. If there really isn't a defined limit, ask a more experienced colleague/friend perhaps, or a professor?

Hope it works out fine.

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    Any journal will give you an upper limit for a manuscript — False. I've never heard of a mathematics journal imposing a page limit. – JeffE Apr 3 '13 at 8:53
  • @JeffE You don't even get a word/character limit? If that truly is the case, I apologize for the generalization. – posdef Apr 3 '13 at 9:41
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    some good qualith math journals have page limits, e.g. Archiv der Mathematik has a soft-ish 10 page limit – Matthew Towers Apr 3 '13 at 13:22
  • @mt_: I stand corrected! – JeffE Apr 4 '13 at 0:54

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