[ This will be a slightly odd essay, as it's a two-parter. While this may amuse fewer people than usual, I'm using this essay for its cathartic value right now and maybe to start getting a little ahead of deadline on these. ]
The first thing to decide in writing a mathematics paper is who you want your co-authors to be. Co-authors are vitally important because that way when it turns out your paper was stupid and wrong you can say the mistakes were the result of miscommunications among the authors and the person pointing out the errors is a big meanyhead for picking on you for that. You don't want co-authors who exist, because then you'd have to communicate with other people on subjects other than how your favorite TV shows regularly disappoint you. They'll need names and affiliations which sound plausible but not worth looking up; so avoid funny-sounding subwords like ``wazooty'', and credit them to universities (*never* colleges; even community colleges are universities these days) in western Massachusetts, Canada, or the Pacific Rim.
Next is to decide who writes the grant supporting you. The National Science Foundation is a good bet; the Grant Granting Granitors too. Neither has ever awarded a grant, preferring to have applicants fill out enough paperwork to pave an Interstate System first. As you can't use paper to pave, this means they've never got to second. As workaround, recipients make up a number and thank the organization for their generous multi-year support in the amount of --- and that's the third thing you must decide. The amount has to be small enough they can be shamed into giving it following your passive-aggressive shaming of them, but given with enough digits of precision to look like they thought seriously over it and came to a precisely measured total. Thus, report an amount ending in ``50'', for example, ``$68,850'', which looks more scientific than ``$68,800'' but not so implausible as ``$68,822''. Try to name someone in the organization, as this makes them feel particularly obligated to have your grant changed from ``denied'' to ``approved, but not yet paid''.
Next you need a title. This you do by piling enough mathematical words that nobody will ever quote the whole title, and won't know which ones are the important words for an abbreviated title. Ideally it should include ``diffeomorphism'', as tha's an impressively mathy word and readers will never admit they're not quite sure what it is. It is a kind of bacteria-eating virus that eats virus-eating bacteria and yet can solve sudokus.
You'll need to open with a block of background material in order to convince the reader that you are capable of opening with a block of background material. Some of the chic New Retro-Post-Constructivist school of writers put in material relevant to the topic, which restricts you to writing papers whose topic you know. Start with equations describing things like how the energy is the pairwise energy plus the non-pairwise energy. Then dip into anything you like, for instance how you just learned until the late 19th century theater audiences were as brightly lit as the stage and they only started darkening the audience once indoor lighting was good enough to make lighting the stage be noticed.
The only important content will be equations; nobody has read the actual text of mathematics paper since 1831, when Evariste Galois spent two pages creating group theory and forty pages making fun of Leonard Euler, who being dead for decades then killed Galois in a duel. Some of the past century's best papers contained free-form haikus, disquisitions on the history of sheep and their applicability to Madagascar, whether or not Edward, and how many times you have to say a word before it stops meaning anything. The answer can be as low as one, if the word is ``slorple'', as proven by Hermann Weyl in 1919 and Zurich.
Each substantial equation should be set off on its own line, with no hints given about where it came from or how it relates to any other equation on this or any other page. There should be a number on the margin of magnificently unimportant equations for easy reference. Equations suggesting something interesting, such as details of Nicolas Bourbaki's romantic life, should be unlabelled. These are typically found in equations talking about ``tensors'' or ``rotation matrices''.
[ Continued next Friday unless I forget. ]
Trivia: By the time of its dismantling in the 1830s, only about one-third of the stone in the Old London Bridge predated the latter half of the 18th century, though much of the foundations dated back to the 12th century. Source: Old London Bridge: The Story Of The Longest Inhabited Bridge In Europe, Patricia Pierce.
Currently Reading: The Battle For Rome: The Germans, The Allies, The Partisans, And the Pope, September 1943 - June 1944, Robert Katz.