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Expository Papers

I've tried to stick to ideas mostly unique to mathematical writing in this entry, but you shouldn't take the omission of more generally applicable guidelines of style and process to mean that they're not important here. It's still necessary to use proper grammar and punctuation, write with concision and clarity, write to an audience, and all that jazz.

## Usage, Style, and Formatting

**Theorems, lemmas, propositions, corollaries, and hypotheses**: What's the difference? It depends somewhat on context, but theorems are generally of major results of independent interest and involve nontrivial proof. A lemma is a smaller result that may have an easy or difficult proof, but is only a stepping stone in the proof of a theorem. The term proposition is not as common, but can refer to a minor theorem. Depending on the audience, the application of these terms can vary. Something that may be stated as a theorem when writing for high school students might be simply a given when addressing professional mathematicians. A corollary is a result that follows easily from a theorem, lemma, or proposition, and a hypothesis is a statement that can form the basis for a proof.

**Using examples**: It is often useful to discuss specific examples before the general case. By beginning with showing how a concept works in a simple case, the author can prepare readers for additional complications that eventually lead to the generally applicable concept.

**Definitions**: Don't overwhelm the reader with definitions, and try to place them close to where the information is being used. In definitions, it is convention to assume that "if" means "if and only if," and to make it clear that a definition is being given, the word being defined is often italicized.

**Notation**: When choosing variable names, make sure that you don't pick letters that already refer to something else by convention. Use variables that are easily distinguishable.

**Words vs. symbols**: When possible, use words instead of symbols so that the reader can more easily follow the writing. Don't thrown in extra symbols where unnecessary (such as naming a matrix when making a simple statement about matrices in general). There are appropriate uses of symbols, however, so use your judgment as to which makes the writing more concise in a given situation.

**Placement and punctuation of equations and expressions**: If an equation is complicated, awkward to put in-line with text, or needs to be numbered for later reference, display it on its own centered line. Mathematical expressions (even when placed on their own line) should be incorporated into sentences and punctuated properly. Don't begin sentences with mathematical expressions.

**Choosing an article**: Be careful not to use "the" to refer a result that might not be unique. For example, don't say that "The solution to the equation is…" if there is potentially more than one solution.

**Passive vs. active voice**: Use active voice when possible. A process or a concept can perform an action, as in "A trigonometric substitution results in…" It is also common to use "we" in explaining the steps taken, as in "We apply so-and-so's theorem to obtain…"

**Format and typesetting**: LATEX is a typesetting language that is commonly used when writing technical papers. It allows for consistent and legible formatting of equations, as well as the incorporation of graphs and figures.

## Types of Writing

**Papers**: The title should carefully balance the goals of communicating the significance of the paper and grabbing the reader's attention with the right amount of detail. Paper's should be dated. An abstract should come at the beginning of the paper (well, after the title and author) and give a concise summary of the key points in the paper, in a way that can be understood by someone even without reading the paper. The introduction should be fairly short and not begin with and overwhelming amount of symbols and complicated notation. Tables should be as simple as possible, and should be numbered and contain a caption. It is useful to consider what numbers in a table will need to be compared, in order to arrange the tables in a logical way. A conclusion section isn't always necessary, but can include a description of noter viewpoint of the results of the work, discuss limitations, or suggest future research. The two common citation styles are by number or the Harvard system (name and year); the important thing is that citations don't hinder the flow of a sentence.

**Proofs**: Readers of proofs want to be able to quickly grasp the process followed and to see how that process could be applicable in other situations. Some readers will also want to know the details of the proof, so it is important to provide enough information about the key methods involved in each step, as well as the relative difficulty of each step. The goal is to be concise and to follow a clearly defined and logical structure to arrive at the result. Proofs often end with the abbreviation QED (which stands for the Latin quod erat demonstrandum), or a tombstone or halmos (a little square).

**Posters**: The goal of a poster is to generate interest and discussion. The poster should briefly outline the work being presented, but should not go into excessive detail. The title does not necessarily have to be the same as that of the paper presented, but should grab the attention of people walking by. It might be helpful to think of the poster as telling a story: It needs to draw the audience in, lead them in an engaging way through the important events, and give a brief but clear conclusion. It is important to remember that the presenter is standing next to the poster to elaborate on details, so that the poster may consist of very brief - even fragmentary - bits of information that the presenter can explain further to those who stop to talk.

Much of this information was taken from Handbook of Writing for the Mathematical Sciences by Nicholas J. Higham (1998). This book has lots more detailed information that you should check out if you want even more mathematical writing tips!