Exam (elaborations) HOW TO WRITE MATHEMATICAL PAPERS

One of the most common shortcomings of both inexperienced and experienced writers is the failure to inform readers of their reasoning. Often, such failures arise when one writes down a long series of equalities without explaining those previous theorems, lemmas, equations, etc. that were used in constructing this long string. Tag equations! Refer to the tagged equations. Tell readers your reasoning at every step. You know all the steps; all is logical to you. But how long did it take you to develop this proof? You cannot expect readers to come to the same reasoning in a much shorter period of time without being aided along the way. Every step (except trivial ones) should be documented. The steps that you undergo should be related in sentences. Don’t just add a parenthetical remark, indicating that a certain equation was used. This is lazy writing. It is also bad grammar. Moreover, aesthetically, such writing is ugly. Parenthetical remarks after equations often cause texing problems. Such lazy composition causes difficulties for copy editors and compositors. Display complicated expressions and equations, for if you put them in the text, they will look like a huge mess. Your paper should be attractive, neat, and beautiful. An attractive paper puts the editor and referee in a good mood. You want the editor to exclaim, “Ah, this paper looks nice! We should publish it.” Putting the editor or referee in a positive psychological mood is important in convincing them of the worth of your paper. Do not use LHS, RHS, etc.; you are not at the blackboard. If you need an abbreviation, it is much better to write: Let S(x, m, n) denote the left-hand side of (3.2.1729). If you use an abbreviation, always explain the abbreviation at its first appearance. Number equations, as well as theorems, corollaries, lemmas, etc. by section. Subdividing your paper into sections enables readers to more easily locate equations and theorems. For example, if an author writes, “Recall that in Theorem 31, we proved . . . ,” it might take the reader nearly a half or full minute to find the theorem, but if the author writes, “Recall that in Theorem 4.3, we proved . . . ,” it is likely that only a few seconds are needed for a search. Use standard notation. Although you may think that your notation is better than existing notation, swallow your pride and avoid your new, spiffy, thoughtful notation. For example, in the mid-twentieth century, a book entitled, The Elliptic Functions as They Should Be, was published. The author, Albert Eagle, desired to introduce a more organized, descriptive, and useful notation to replace the current notation in the theory of elliptic functions. His book was a flop; no one has ever used his notation. Notation evolves over a long period, and it is very difficult to “change horses in midstream.” Many mathematicians have usually contributed to the notation in any particular field. Use as little notation as possible. Think about which subscripts, superscripts, additional arguments, and fonts would be best. Try to avoid subscripts of subscripts. Clarity and familiarity of notation aid the readers in her or his understanding of what you are trying to communicate. As an editor, I have read many reports from referees, and one of the most common complaints of referees is that authors forget to give definitions. It is very easy to use concepts and terminology that you have not defined. Have you defined everything that you need to define? Have you explained all the notation that you have used?